Let be a compact
-manifold with nonempty
boundary and let
be a boundary-preserving
map. Denote by
the minimum number of fixed point among all boundary-preserving
maps that are homotopic through boundary-preserving maps to
. The relative
Nielsen number
is the sum of the number of essential fixed point classes of the restriction
and the number of essential fixed point classes of
that do not contain essential fixed point classes of
. We prove
that if
is the Möbius band with one (open) disc removed, then
for all maps
. This result
is the final step in the boundary-Wecken classification of surfaces, which is as follows. If
is the disc, annulus or
Möbius band, then is
boundary-Wecken, that is, for
all boundary-preserving maps. If
is the disc with two discs removed or the Möbius band with one disc removed, then
is not
boundary-Wecken, but .
All other surfaces are totally non-boundary-Wecken, that is, given an integer
, there
is a map
such that .